RF Circuit Design - [Ch3-1] Microwave Network

Chapter 3-1

Scattering Matrix and

Microwave Network

Chien-Jung Li

Department of Electronics Engineering

National Taipei University of Technology

Department of Electronic Engineering, NTUT

Traveling Waves

j xV x Ae j xV x Be

j x j xV x V x V x Ae Be

0 0

V x V xI x I x I x

Z Z

V xx

V x

• Introducing the notation of the voltage and current traveling

waves:

and

• The reflection coefficient between incident and reflected

wave can be written as:

2/32


Normalized Traveling Waves

0

0

1 1

2 2b x v x i x V x Z I x

Z

0

0

1 1

2 2a x v x i x V x Z I x

Z

• Normalized notation of voltage and current waves:

0

V xv x

Z 0i x Z I x

0

V xa x

Z

0

V xb x

Z v x a x b x

i x a x b x

b x x a x

Normalized incident wave

Normalized reflected wave

and

Introduce normalization to

relate voltage with power.

2

2

0

V xa x

Z

210log 10log 20logaP a x a x

3/32


Two-port Network

Two-port

Network

2 2a l

2 2b l

2 2a x

2 2b x

1 1a l

1 1b l

1 1a x

1 1b x

1oZ 2oZ

Input port Output port

Port 1

1 1x lPort 2

2 2x l

• If instead of a one-port transmission line we have the two-port network

shown with incident wave and reflected wave at port 1

(located at ), and incident wave and reflected wave

1 1a l 1 1b l

1 1x l 2 2a l

2 2b l 2 2x l at port 2 (located at )

At port 1

Reflected wave

Incident wave 1 1a l

1 1b l

At port 1

Reflected wave

Incident wave 2 2a l

2 2b l

4/32


Scattering Matrix (I)

1 1 11 1 1 12 2 2b S a S al l l

2 2 21 1 1 22 2 2b S a S al l l

1 1 1 111 12

2 2 2 221 22

b aS S

b aS S

l l

l l

Scattering matrix Scattering parameters

Two-port

Network

2 2a l

2 2b l

2 2a x

2 2b x

1 1a l

1 1b l

1 1a x

1 1b x

1oZ 2oZ


Port 1

1 1x lPort 2

2 2x l

incident to the ports reflected from the ports

5/32


Scattering Matrix (II)

contribution to the reflected wave 1 1b l

due to incident wave 2 2a l at port 2

2 2 21 1 1 22 2 2b S a S al l l

1 1 11 1 1 12 2 2b S a S al l l




due to incident wave 2 2a l at port 2 contribution to the reflected wave 2 2b l


Two-port

Network

2 2a l

2 2b l

2 2a x

2 2b x

1 1a l

1 1b l

1 1a x

1 1b x

1oZ 2oZ


Port 1

1 1x lPort 2

2 2x l

6/32


Scattering Parameters

2 2

1 1

11

1 1 0a

bS

al

l

lInput reflection coefficient with output properly terminated

1 1

2 2

22

2 2 0a

bS

al

l

lOutput reflection coefficient with input properly terminated

2 2

2 2

21

1 1 0a

bS

al

l

lForward transmission coefficient with output properly terminated

1 1

1 1

12

2 2 0a

bS

al

l

lReverse transmission coefficient with output properly terminated

(measured with port 2 properly terminated)




7/32


Return Loss and Insertion Loss

2 2

1 1

11

1 1 0a

bS

al

l

l

• Return Loss (RL)

2 2

2 2

21

1 1 0a

bS

al

l

l

2

2 1 1 111 2

1 1 1

b

a

b PS

a P

l

l 21

11 11

1

10log 10log 20log (dB)b

a

PS S

P

11Return Loss (RL) 10log 20log (dB)in

reft

PS

P

(折返損耗, 反射損耗)

2

2 2 2 221 2

1 1 1

b

a

b PS

a P

l

l 22

21 21

1

10log 10log 20log (dB)b

a

PS S

P

21Insertion Loss (IL) 10log 20log (dB)transmit

receive

PS

P

(植入損耗, 插入損耗) • Insertion Loss (IL)

8/32


Procedure of Measuring S11

Two-port

Network

2 2 0a l

2 2b l

1 1a l

1 1b l1oZ 2oZ

Port 1

1 1x lPort 2

2 2x l

2 2oZ Z

1E

1 1oZ Z

2 2

1 1

11

1 1 0a

bS

al

l

lOUTZ

• With Z2=Zo2 the condition is satisfied. Similar considerations

apply to measurements at the input port. Also the characteristic

impedances of the transmission lines are usually identical (i.e. Zo1=Zo2),

with a 50 Ω being the standard value.

2 2 0a l

matched

2 2

1 1

11

1 1 0a

bS

al

l

l 1 1 11 1 1 12 2 2b S a S al l l

0

9/32


n-port Network (I)

• The transmission lines are assumed

to be lossless with characteristic

impedance Zoi (i=1 to n). The

scattering matrix of the n port, at

the unprimed reference planes, in

the form

b S a

n-port

Network

1oZ

Port 1 Port 1'

1TZ

1 1a l

1 1b l

2oZ

Port 2 Port 2'

2 2a l

2 2b l

onZ

Port n Port n'

n na l

n nb l

1 21

2o oa Z V Z I

1 21

2o ob Z V Z I

11 12 1

21 22 2

1 2

n

n

n n nn

S S S

S S S

S

S S S

1 2

1

1 2

2

1 2

1 2

0 0

0 0

0 0

o

o

o

on

Z

Z

Z

Z

10/32


n-port Network (II)

• The [a], [b], [V], and [I] are column matrices. That is

1

2

n

a

a

a

a

1

2

n

b

b

b

b

1

2

n

V

V

V

V

1

2

n

I

I

I

I

l

l

1 1 1 111

1 1 1 10 2,3, ,j

T o

T oa j n

b Z ZS

a Z Z

The S parameters of the n-port networks are easily measured.

For example S11 at x1=l1 is given by

where ZT1 is the impedance seen at port 1 with the other ports matched.

11/32


Reference Planes

• In practice, we often need to attach transmission lines to

the network under test for the measurement. Since the S

parameters are measured using traveling waves, we need

to specify the positions where the measurements are

made.

Device Under Test

(DUT)

1l

2l

Unprimed reference plane

12/32


Shifting the Reference Planes

1 1 1 111 12

2 2 2 221 22

b aS S

b aS S

l l

l l

1 111 12

2 221 22

0 0

0 0

b aS S

b aS S

• At port 1 and port 2

• At port 1' and port 2'

The angles and are the electrical lengths of the transmission line

between the primed and unprimed reference planes.

Two-port Network

2 2a l

2 2b l

1 1a l

1 1b l1oZ 2oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

1 0a

1 0b

Port 2'

2 0x

2 0a

2 0b

2 2l 1 1l

Primed reference plane

1 2

Unprimed reference plane

11 12

21 22

S S

S S

13/32


Shifting the Reference Planes

1

1 1 1 0 jb b el

1

1 1 1 0 ja a el

2

2 2 2 0 jb b el

2

2 2 2 0 ja a el

1 21

1 2 2

2

1 1 111 12 11 12

22 2 221 22 21 22

0 0 0

0 0 0

jj

j j

b a aS S S e S e

b a aS S S e S e

1 21

1 2 2

2

11 12 11 12

221 22 21 22

jj

j j

S S S e S e

S S S e S e

1 21

1 2 2

2

11 12 11 12

221 22 21 22

jj

j j

S S S e S e

S S S e S e

2 2a l

2 2b l

1 1a l

1 1b l1oZ 2oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

1 0a

1 0b

Port 2'

2 0x

2 0a

2 0b

2 2l 1 1l

Reference planes

11 12

21 22

S S

S S

14/32


Properties of Scattering Parameters

• In order to know the properties of scattering parameters, let’s start

with a two-port network that has two transmission lines attached at

its input and output terminals. (Without considering the source and

load)

1oZ 2oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

1 1I x

Port 2'

2 0x

2l1l

1 1V x

2 2I x

2 2V x

0iP

0iP

0iP

0iP

• Find the incident power and reflected power . (i=1 for port 1 and i=2 for port 2)

0iP 0iP

11 12

21 22

S S

S S

15/32


Incident and Reflected Power

• Average power of incident wave on the primed ith port (x1=0, x2=0)

2

2 2

,

01 1 10 Re 0 0 0 0

2 2 2

i

i i i i i rms

oi

VP V I a a

Z

21

0 02

i iP a

2

2 2

,

01 1 10 Re 0 0 0 0

2 2 2

i

i i i i i rms

oi

VP V I b b

Z

• Average reflected power

• Since the transmission lines are assumed to be lossless, we have

0i i iP P l

0i i iP P l

2 21 1

02 2

i i ia a x

2 21 1

02 2

i i ib b x

No power loss everywhere on the lines

21

0 02

i iP b

16/32


Consider Matched Source and Load (I)

2 2 20 0oV Z I

2 2 2 2 2 2 2 2

2 2

1 10 0 0 0 0 0

2 2o o o

o o

a V Z I Z I Z IZ Z

It follows that

Two-port

Network

1oZPort 1

1 1x lPort 2

2 2x lPort 1'

1 0x

1 0I

Port 2'

2 0x

2l1l

1 0V

2 0I

2 2V l

2oZ

1 1I l 2 2I l

1 1V l

1TZ

2 2a x

2 2b x

1 1a x

1 1b x

2 0V

1E

1 1oZ Z

2 2oZ Z

matched

matched

No reflection from load • At x2=0, we have

1

1 1 1 1

1 1

10 0 0

2 2o

o o

Ea V Z I

Z Z

22 1

1

1

04 o

Ea

Z

1 1 1 10 0oV E Z I• At x1=0, we have

It follows that and

Vpp

17/32


Consider Matched Source and Load (II)

22 1

1 1

1

10 0

2 8AVS

o

EP P a

Z

• Since the line is lossless, we have

2 2

1 1 1

1 10

2 2a a l

Power available from the source is independent of

the input impedance ZT1 of the two-port network

• The power available from the source E1 with internal resistance

Z1=Zo1 is equal to the power of incident wave at x1=0:

The available power PAVS is the incident power at x1=0.

18/32


Mismatched Source (I)

2 1 1 1 1 1 1

1

1

0 0 0 010

2 8

o o

o

V Z I V Z Ia

Z

2 22

1 1 1 1 1 1 1 1 1

1

10 0 0 0 0 0

8o o o

o

V Z I V Z V I Z IZ

2 2 22

1 1 1 1 1 1 1 1 1 1

1

1 10 0 0 0 0 0 0

2 8o o o

o

b V Z I V Z I V Z IZ

2 2

1 1 1 1 1 1 1

1 1 10 0 0 0 0 0 0

2 2 4P a b I V I V

1 1

1Re 0 0

2I V

• Consider that if Z1 is not equal to Zo1

Similarly,

• Power delivered to port 1', or to port 1 (since the line is lossless) is

19/32


Mismatched Source (II)

2

1 1

10 0

2AVSb P P

l l 2

1 1 1 1

1

2AVSb P P

l 2

1 1 1 1

10 0

2AVSP P P b

2 2 2 2 2 2 2 2 2 2

2 2

1 10 0 0 0 0 0

2 2o o o o

o o

b V Z I Z I Z I Z IZ Z

• Reflected power from port 1 (or port 1')

It can also be written as

• If ZT1=Zo1, then the reflected power is zero. However, if ZT1≠Zo1, part of

the incident power is reflected back to the generator. The net power

delivered to port 1 is

We can obtain

20/32


Calculation of S11 and S21 (I)

2 2

2 2 2 2

1 10 0 0

2 2oP b I Z

l l

l l

l l

2 2 2 2

1 1 1 1

11

1 1 1 10 0a V

b VS

a V

1 111

1 1

T o

T o

Z ZS

Z Z

• Power delivered to the load Z2 (=Zo2)

• Calculate the S-parameter

S11 is the reflection coefficient of port 1 with port 2 terminated in its

normalizing impedance Zo2. (a2=0)

• The evaluation of S11 at x1=0 (S'11) can be done using .

Alternately, we can calculate the input impedance at x1=0, and its

associated reflection coefficient would be S'11.

12

11 11

jS S e l

21/32


Calculation of S11 and S21 (II)

l

l l

l

2 2

2

2 1 1 1 1

11 2

1 10

AVS

AVSa

b P PS

Pa

l 2

1 1 1 110 1AVSP P P S

l l

l l

l l

2 2 2 2

2 2 2 2 1

21

1 1 1 1 10 0

o

oa I

b Z IS

a Z I

l

l

l

2 2

2 2 2

1 1 1 0

o

o I

Z I

Z I

• The ratio of the power reflected from port 1 to the power available at

port 1.

or

• If , the power reflected is larger than the power available at

port 1. In this case, port 1 acts as a source of power and oscillations

can occur.

• Evaluation of S21 at unprimed reference plane

2 2 2 2 2 2 2 2 2 2 since 0I I I I Il l l l l

11 1S

22/32


Find S Parameter by Excitation (I)

• Thevenin’s equivalent network

1

1, 1

j

THE E e l

Two-port

Network

Port 1

1 1x lPort 2

2 2x l

2 2V l

1 1I l 2 2I l

1 1V l

1TZ

2 2 0a l

2 2b l

1 1a l

1 1b l

1,THE

1oZ

2oZ

Two-port

Network

1oZPort 1

1 1x lPort 2

2 2x lPort 1'

1 0x

1 0I

Port 2'

2 0x

2l1l

1 0V

2 0I

2 2V l

2oZ

1 1I l 2 2I l

1 1V l

1TZ

2 2a x

2 2b x

1 1a x

1 1b x

2 0V

1E

1 1oZ Z

2 2oZ Z

matched

matched

23/32


Find S Parameter by Excitation (II)

l

l l l 1 1

1 1 1 1 1 1 1

11

1

2o

oo

aI V Z I

ZZ


Two-port

Network

Port 1

1 1x lPort 2

2 2x l

2 2V l

1 1I l 2 2I l

1 1V l

1TZ

2 2 0a l

2 2b l

1 1a l

1 1b l

1,THE

1oZ

2oZ

l l 1 1 1, 1 1 1TH oV E Z I

l 1,

1 1

12

TH

o

EI

Z

l

l 2 2

2 2

2o

VI

Z

2 2

2 2 2 2 21

21

1,1 1 1 20

2o o

THo oI

Z I VZS

EZ I Z

l

l l

l

• At port 1:

• At port 2:

• The S21:

S21 represents a forward voltage

transmission coefficient from

port 1 to port 2.

24/32


Find S Parameter by Excitation (III)

2

2 2

22212

1,

1

1

2

8

oT

TH

o

V

ZG S

E

Z

l

221

1,

2

TH

VS

E

2

2 221

1, 2L

T

AVS TH

P VG S

P E

GT represents the ratio of the power delivered to the load Zo2 (i.e., PL) to the

power available from the source E1,TH (i.e., PAVS).

• If Z1 = Z2 = Zo

• Transducer Power Gain: 2

21TG S

and

25/32


Find S Parameter by Excitation (IV)

1 1

2 2 2 222

2 2 2 20

T o

T oa

b Z ZS

a Z Zl

l

l

1 1

1 1 2 1 1

12

2 2 1 2,0

2o

o THa

b Z VS

a Z El

l l

l

• Excitation at port2’ by E2 with source impedance Z2=Zo2 is placed at port 2’ and

port 1’ is matched (Z1=Zo1) we find that at the unprimed reference planes

S22 is the reflection coefficient of port 2 with port 1 terminated in its

normalizing impedance Z1= Zo1. (a1(l1)=0) , and S12 represents a reverse

voltage evaluate S’22 and S’12 at the primed reference planes.

Two-port

Network 1oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

Port 2'

2 0x

2l1l

2oZ

2TZ

2 2a l

2 2b l

1 1 0a l

1 1b l

2E

2 2oZ Z

1 1oZ Z

26/32


Find S Parameter by Excitation (V)

2

1 1

2 112 2

2,

2

1

2

8

o

TH

o

V

ZS

E

Z

l

The S parameter of a transistor are commonly ,measured with Zo1=Zo2=Zo

and Z1 = Z2 = Zo. These S parameters are said to be measured in a Zo system.

If this transistor is then used in the circuit with arbitrary terminations Z1 and

Z2, the gain GT is no longer as given. GT can be expressed in terms of Z1, Z2,

and the S parameters of the transistor measured in a Zo system.

• Reverse Transducer Power Gain:

27/32


Example – S Parameter of a Series Z (I)

• Evaluate the S parameters, in a Zo system, of a series impedance Z.

Z

oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

Port 2'

2 0x

oZ

1TZ

2 2a l

2 2b l

1 1a l

1 1b l

1E

1 oZ Z

2 oZ Z

matched

matched

Z

Port 1 Port 2

28/32


Example – S Parameter of a Series Z (II)


2 2

1 1 111

1 1 10

T o

T oa

b Z ZS

a Z Z

l

l

l1T oZ Z Z where 11

2 o

ZS

Z Z

Z

Port 1 Port 2

1TZ

1 1a l

1 1b l

1,THE

oZ

oZ

2 2V l

2 2 1,2o

TH

o

ZV E

Z Z

l

1,2 2

21

1, 1,

2 2

22 2

oTH

o o

oTH TH

ZE

V Z Z ZS

Z ZE E

l

(1)

(2)

For symmetry, we observe that S22=S11 and S12 =S21. (Reciprocal condition)

• For , and in a system: 100 Z j 50 0.707 45 0.707 45

0.707 45 0.707 45S

29/32


Example – S Parameter of a Shunt Y (I)

• Evaluate the S parameters, in a Zo system, of a shunt admittance Y.

oZ

Port 1

1 1x lPort 2

2 2x lPort 1'

1 0x

Port 2'

2 0x

oZ

1TZ

2 2a l

2 2b l

1 1a l

1 1b l

1E

1 oZ Z

2 oZ Z

matched

matched

Y

Port 1 Port 2

Y

30/32


Example – S Parameter of a Shunt Y (II)


2 2

1 1 111

1 1 10

T o

T oa

b Z ZS

a Z Z

l

l

l1

1||

1o

T o

o

ZZ Z

Y Z Y

where 11

2o

o

Z YS

Z Y

Port 1 Port 2

1TZ

1 1a l

1 1b l

1,THE

oZ

oZ

2 2V l

1,12 2 1,

1 2

THTTH

T o o

EZV E

Z Z Z Y

l

2 2

21

1,

2

22 oTH

VS

Z YE

l

(1)

(2)

For symmetry, we observe that S22=S11 and S12 =S21. (Reciprocal condition)

• For , and in a system: 10 mSY 50 0.2 0.8

0.8 0.2S

Y

31/32


Summary

• For a 2-port network: 11 12

21 22

S SS

S S

l 2

1 1 1 110 1AVSP P P S

21

0 02

i iP a

21

0 02

i iP b

• Average incident power:

• Average reflected power:

With lossless lines:

2 21 1

0 02 2

i i i i i iP P x a a x

2 21 1

0 02 2

i i i i i iP P x b b x

2

1 1

10 0

2AVSP P a • Available power from source:

With matched condition:

22 1

1 1

1

10 0

2 8AVS

o

EP P a

Z

2 2 2

1 1 1 1

1 1 10 0 0 0

2 2 2AVSP a b P b • Power delivered to port 1:

With lossless lines:

32/32